3.681 \(\int \frac{\sqrt{c+d x^2}}{x \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=80 \[ \frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a \sqrt{b}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a} \]

[Out]

-((Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a) + (Sqrt[b*c - a*d]*ArcTanh[(Sqrt
[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b])

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Rubi [A]  time = 0.207051, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\sqrt{b c-a d} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{a \sqrt{b}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c+d x^2}}{\sqrt{c}}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c + d*x^2]/(x*(a + b*x^2)),x]

[Out]

-((Sqrt[c]*ArcTanh[Sqrt[c + d*x^2]/Sqrt[c]])/a) + (Sqrt[b*c - a*d]*ArcTanh[(Sqrt
[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/(a*Sqrt[b])

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Rubi in Sympy [A]  time = 26.017, size = 66, normalized size = 0.82 \[ - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c + d x^{2}}}{\sqrt{c}} \right )}}{a} + \frac{\sqrt{a d - b c} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{a \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x**2+c)**(1/2)/x/(b*x**2+a),x)

[Out]

-sqrt(c)*atanh(sqrt(c + d*x**2)/sqrt(c))/a + sqrt(a*d - b*c)*atan(sqrt(b)*sqrt(c
+ d*x**2)/sqrt(a*d - b*c))/(a*sqrt(b))

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Mathematica [C]  time = 0.602429, size = 229, normalized size = 2.86 \[ \frac{\frac{\sqrt{b c-a d} \left (\log \left (-\frac{2 a \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x+i \sqrt{a}\right ) (b c-a d)^{3/2}}\right )+\log \left (-\frac{2 a \sqrt{b} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (\sqrt{b} x-i \sqrt{a}\right ) (b c-a d)^{3/2}}\right )\right )}{\sqrt{b}}-2 \sqrt{c} \log \left (\sqrt{c} \sqrt{c+d x^2}+c\right )+2 \sqrt{c} \log (x)}{2 a} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c + d*x^2]/(x*(a + b*x^2)),x]

[Out]

(2*Sqrt[c]*Log[x] - 2*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c + d*x^2]] + (Sqrt[b*c - a*d
]*(Log[(-2*a*Sqrt[b]*(Sqrt[b]*c - I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2
]))/((b*c - a*d)^(3/2)*(I*Sqrt[a] + Sqrt[b]*x))] + Log[(-2*a*Sqrt[b]*(Sqrt[b]*c
+ I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))/((b*c - a*d)^(3/2)*((-I)*Sqr
t[a] + Sqrt[b]*x))]))/Sqrt[b])/(2*a)

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Maple [B]  time = 0.017, size = 984, normalized size = 12.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x^2+c)^(1/2)/x/(b*x^2+a),x)

[Out]

-1/a*c^(1/2)*ln((2*c+2*c^(1/2)*(d*x^2+c)^(1/2))/x)+1/a*(d*x^2+c)^(1/2)-1/2/a*((x
-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2
)-1/2/a*d^(1/2)*(-a*b)^(1/2)/b*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1
/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/
b)^(1/2))-1/2/b/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/
b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2
)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d+1/2/a/(-(a*
d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(
a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/
2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*c-1/2/a*((x+1/b*(-a*b)^(1/2))^2*d-
2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)+1/2/a*d^(1/2)*(-a*b)^
(1/2)/b*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/
2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/2/b/(-(a*d
-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a
*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2
))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d+1/2/a/(-(a*d-b*c)/b)^(1/2)*ln((-2
*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+
1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)
)/(x+1/b*(-a*b)^(1/2)))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{d x^{2} + c}}{{\left (b x^{2} + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x),x, algorithm="maxima")

[Out]

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x), x)

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Fricas [A]  time = 0.317225, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 2 \, \sqrt{c} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right )}{4 \, a}, -\frac{4 \, \sqrt{-c} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) - \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, a}, \frac{\sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) + \sqrt{c} \log \left (-\frac{d x^{2} - 2 \, \sqrt{d x^{2} + c} \sqrt{c} + 2 \, c}{x^{2}}\right )}{2 \, a}, -\frac{2 \, \sqrt{-c} \arctan \left (\frac{c}{\sqrt{d x^{2} + c} \sqrt{-c}}\right ) - \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x),x, algorithm="fricas")

[Out]

[1/4*(sqrt((b*c - a*d)/b)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2
*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*s
qrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)) + 2*sqrt(c)*log(-(d*x^2 - 2*sqr
t(d*x^2 + c)*sqrt(c) + 2*c)/x^2))/a, -1/4*(4*sqrt(-c)*arctan(c/(sqrt(d*x^2 + c)*
sqrt(-c))) - sqrt((b*c - a*d)/b)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*
d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2
+ c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b*x^2 + a^2)))/a, 1/2*(sqrt(-(b*c - a*
d)/b)*arctan(1/2*(b*d*x^2 + 2*b*c - a*d)/(sqrt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b)
)) + sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2))/a, -1/2*(2*sqr
t(-c)*arctan(c/(sqrt(d*x^2 + c)*sqrt(-c))) - sqrt(-(b*c - a*d)/b)*arctan(1/2*(b*
d*x^2 + 2*b*c - a*d)/(sqrt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b))))/a]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c + d x^{2}}}{x \left (a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x**2+c)**(1/2)/x/(b*x**2+a),x)

[Out]

Integral(sqrt(c + d*x**2)/(x*(a + b*x**2)), x)

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GIAC/XCAS [A]  time = 0.232818, size = 117, normalized size = 1.46 \[ -d{\left (\frac{{\left (b c - a d\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} a d} - \frac{c \arctan \left (\frac{\sqrt{d x^{2} + c}}{\sqrt{-c}}\right )}{a \sqrt{-c} d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x),x, algorithm="giac")

[Out]

-d*((b*c - a*d)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*
b*d)*a*d) - c*arctan(sqrt(d*x^2 + c)/sqrt(-c))/(a*sqrt(-c)*d))